The corner of my lip is super dry and chapped. It hurts to open my mouth I think maybe it's a cold sore but I'm not sure. How do I make it go away?

Angular Cheilitis. There is a condition called cheilitis where lesions develop in the corners of the mouth and they can be persistent. They can be from lip-licking, nutrient deficiencies, dry mouth etc. These can become subject to opportunistic infections. Using a topical antibiotic and discouraging harmful behaviors (lip licking) may help.